Source: http://www.abarim-publications.com/Dictionary/d/d-y.html

די

There appear to be two different Hebrew particles די (dy), which don't have anything to do with each other. A third, Aramaicדי (dy), meaning 'which' appears lavishly in Ezra and Daniel.

די I

The substantive די (day) appears to be unique to the Hebrew language. It means sufficiently, enough or even abundance. It frequently occurs in constructions with other words (Esther 1:18, 2 Chronicles 30:3, Malachi 3:10 only) and often combined with prefixes:

With ב (be), meaning in: בדי (beday), meaning what suffices for, or for the need of (Nahum 2:13, Jeremiah 51:58) or in the abundance of (Job 39:25), or according to the sufficiency (Leviticus 25:26, Deuteronomy 25:2) or according to the abundance of (Judges 6:5).

With מ (me), meaning from or out of: מדי (meday), meaning out of the abundance of, or as often as (1 Samuel 1:7, 1 Kings 14:28, Jeremiah 31:20).

די II

A second, identical particle occurs frequently in combination with the preposition עם ('im), meaning with, to form the term עמדי, commonly translated with 'with me'. How this די managed to acquire the meaning of 'me' isn't known (merely a postfixed letter י, yod, means 'me'), but something similar appears in Arabic (says BDB Theological Dictionary).